G-spectrum
In algebraic topology, a G-spectrum is a spectrum with an action of a (finite) group. Let X be a spectrum with an action of a finite group G. The important notion is that of the homotopy fixed point set [math]\displaystyle{ X^{hG} }[/math]. There is always
- [math]\displaystyle{ X^G \to X^{hG}, }[/math]
a map from the fixed point spectrum to a homotopy fixed point spectrum (because, by definition, [math]\displaystyle{ X^{hG} }[/math] is the mapping spectrum [math]\displaystyle{ F(BG_+, X)^G }[/math].)
Example: [math]\displaystyle{ \mathbb{Z}/2 }[/math] acts on the complex K-theory KU by taking the conjugate bundle of a complex vector bundle. Then [math]\displaystyle{ KU^{h\mathbb{Z}/2} = KO }[/math], the real K-theory.
The cofiber of [math]\displaystyle{ X_{hG} \to X^{hG} }[/math] is called the Tate spectrum of X.
G-Galois extension in the sense of Rognes
This notion is due to J. Rognes (Rognes 2008). Let A be an E∞-ring with an action of a finite group G and B = AhG its invariant subring. Then B → A (the map of B-algebras in E∞-sense) is said to be a G-Galois extension if the natural map
- [math]\displaystyle{ A \otimes_B A \to \prod_{g \in G} A }[/math]
(which generalizes [math]\displaystyle{ x \otimes y \mapsto (g(x) y) }[/math] in the classical setup) is an equivalence. The extension is faithful if the Bousfield classes of A, B over B are equivalent.
Example: KO → KU is a [math]\displaystyle{ \mathbb{Z} }[/math]./2-Galois extension.
See also
References
- Mathew, Akhil; Meier, Lennart (2015). "Affineness and chromatic homotopy theory". Journal of Topology 8 (2): 476–528. doi:10.1112/jtopol/jtv005.
- Rognes, John (2008), "Galois extensions of structured ring spectra. Stably dualizable groups", Memoirs of the American Mathematical Society 192 (898), doi:10.1090/memo/0898
External links
- "Homology of homotopy fixed point spectra". MathOverflow. June 30, 2012. https://mathoverflow.net/q/101011.
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